[blair]_convex optimization and lagrange multipliers (1977)

8/11/2019 [Blair]_Convex Optimization and Lagrange Multipliers (1977)

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Working

Papers

CONVEX

OPTIMIZATION

AND

LAGRANGE

MULTIPLIERS

Charles

E.

Blair

#407

College of

Commerce and

Business

Administration

University

of

Illinois

at Urbana-Champaign


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FACULTY

WORKING

PAPERS

College

of

Commerce

and

Business

Administration

University

of

Illinois

at

Urbana-Champaign

June

8,

1977

CONVEX

OPTIMIZATION

AND

LAGRANGE

MULTIPLIERS

Charles

E.

Blair

#407


8/28

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Convex

Optimization

and

Lagrange Multipliers

by

Charles

E.

Blair

Dept.

of Business

Administration

June

8,

1977

This work was

supported by a

grant

from

Investors

in

Business Education,

University

of

Illinois.

Abstract

We show how

the duality

theorem of linear

progranmiing

can be used to

prove

several

results

on

general

convex

optimization.


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Let

f,

g,

,...,g,

be

convex

functions

defined

on

a

convex subset

S

of

a vector

space.

Let

T

=

{x^S

|

g

.

(x)_0. If

A

is

non-empty

and every

xA satisfies

dx>e,

then

there

are U>0

and

V

such

that

U

B+V

C=d

and

Ub+Vc>_e.

We will

not

require

separating hyperplane

theorems or

results from

semi-

infinite

programming.

Theorem 1 ; f

(x)^L for

every

xT

if

and only

if for every finite

FCS

there are

X,

,...,A, such

that f

(x)+ZX.g.

(x)^L

for every x^F.

Proof :

The

if part is

immediate.

For

the

only if part it

suffices

to

prove

the result

for

those finite

F

which

contain

members

of

T.

Let

F={y^,...,y

}

y,ST.

For

such

F

the system

of

equations and

inequalities

in

unknowns

9 ,,...,

6,,

1

N

()

E

e.

=

i

1

^

N

E e.g.(y.)^0

llJlk

e.

>

X

has

the

solution

e,=l.

By

convexity,

if

9 .

is a solution

to (D)

,

ES.y

ST.


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.

(.

'

o

-

,^:

I

'H:'-^,.


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Since

we

are

assuming

f(x)>L

for

xT,

every

solution

to (D)

must

satisfy

Ze

f(y.)^L.

By

the

lemma there are X,>^0 and

y

such that

Y

+

S

X

(-g

(y

))

L

for

x^T

and

that

there

is

a

y

for

which

g.(y)^L

for

x^S.

Proof:

Let

&

=

max

{g.(y)}.

For x^S

let

A^={

(X^,

.

.

.

,X|^) |X^>^0,

f(x)+ZX.g.(x)>L,

and

-6(5;X.)L

for

every

xSS if

and only if,

for every

N>0,

there

is

an x^S

such

that

f(x)


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f(y^)

+

E

X

g

(y^)>L

y^SF

3_

J

J

(F;N)

IX^

X

If, for

some

N,

(F;N)

had

a

solution

for

every

finite

FCS,

a

compactness

argument

similar

to that

in

the

corollary

to

Theorem

1

would

yield suitable

multipliers

X,.

Since we

are

assinning such

X.

do

not

exist,

it

must be

that

for

every

N>0

there

is an

F

such that

(F;N)

has

no

solution.

By

the

lemma,

if

(F;N) has no

solution,

there are

8-,

,

.

9j^

2l

^^^

T^O

such

that

M

2

QsA7J

-

yo,

l


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imply

max

g.

(x)>^0.)

Since

the

condition

given

by

Theorem

2

fails for

N=inax

(0,

-^

(h(0)-h(6)),

suitable

X^ exist.) Q.E.D.

Finally,

we

use

a

variation

of

these

techniques

to strengthen a recent

result

of Duff

in

and

Jeroslow

[4].

Theorem

3

:

Let

S=r'^. Assume that

for

X.>0,

f (x)+EX.g. (x)>L,

x6S. Then

there are

affine

functions

h.

(x)=a.x+b

.

(a.SR

,

b.R) such that

h.(x)


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which

contradicts

our

assumption about

the

A..

Therefore

(E) has

solutions

for

every

finite F.

To

complete the

proof

we must

show

0

T^ is

non-empty.

Let

e.=jth

unit

vector.

We

show

that if

F

contains +e.

l


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REFERENCES

1.

Duff

in,

R.J.

Convex

Analysis

Treated

by Linear

Programming.

Mathematical

Programming

,

4,

pp.

125-43.

2.

. The

Lagrange Multiplier

Method for Convex Programming.

Proceedings of National Academy of Sciences

, 72,

pp.

1778-1781.

3.

. Convex

Programming Having

Some Linear

Constraints.

Proceedings

of National

Academy of Sciences

,

74,

pp.

26-8.

4.

Duffin, R.J. and

Jeroslow,

R.G.

Private communication.

5.

Stoer,

J.

and

Witzgall,

C.

Convexity

and

Optimization

in

Finite

Dimens

ions

I

. Springer-Verlag, 1970.


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